Using simulations, a bias caused by inappropriate averaging in experiments with randomized stimuli is described. As an illustrating example, experiments aimed at demonstrating "presentiment" by showing arousal to be higher prior to arousing stimuli than prior to calm stimuli are considered. It is shown that such results could be obtained if (a) the participant believes that the likelihood of an activating stimulus being presented on the next trial increases after a calm stimulus has been presented (the gambler's fallacy) and (b) overall arousal means are calculated across individual arousal means or for a pooled sample of all possible stimulus sequences. The effect becomes very small when ...